Some mathematicians defined this type incomplete-version of Bessel function or this type generalized-version of incomplete gamma function:
K v ( x , y ) = ∫ 1 ∞ e − x t − y t t v + 1 d t {\displaystyle K_{v}(x,y)=\int _{1}^{\infty }{\frac {e^{-xt-{\frac {y}{t}}}}{t^{v+1}}}~dt}
γ ( α , x ; b ) = ∫ 0 x t α − 1 e − t − b t d t {\displaystyle \gamma (\alpha ,x;b)=\int _{0}^{x}t^{\alpha -1}e^{-t-{\frac {b}{t}}}~dt}
Γ ( α , x ; b ) = ∫ x ∞ t α − 1 e − t − b t d t {\displaystyle \Gamma (\alpha ,x;b)=\int _{x}^{\infty }t^{\alpha -1}e^{-t-{\frac {b}{t}}}~dt}
Properties
K v ( x , y ) = x v Γ ( − v , x ; x y ) {\displaystyle K_{v}(x,y)=x^{v}\Gamma (-v,x;xy)}
K v ( x , y ) + K − v ( y , x ) = 2 x v 2 y v 2 K v ( 2 x y ) {\displaystyle K_{v}(x,y)+K_{-v}(y,x)={\frac {2x^{\frac {v}{2}}}{y^{\frac {v}{2}}}}K_{v}(2{\sqrt {xy}})}
γ ( α , x ; 0 ) = γ ( α , x ) {\displaystyle \gamma (\alpha ,x;0)=\gamma (\alpha ,x)}
Γ ( α , x ; 0 ) = Γ ( α , x ) {\displaystyle \Gamma (\alpha ,x;0)=\Gamma (\alpha ,x)}
γ ( α , x ; b ) + Γ ( α , x ; b ) = 2 b α 2 K α ( 2 b ) {\displaystyle \gamma (\alpha ,x;b)+\Gamma (\alpha ,x;b)=2b^{\frac {\alpha }{2}}K_{\alpha }(2{\sqrt {b}})}
One advantage of defining this version of incomplete Bessel function K v ( x , y ) {\displaystyle K_{v}(x,y)} is that for example even the associated Anger–Weber function defined in the Digital Library of Mathematical Functions is related to it:
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